In the primary recovery of oil from a subterranean, oil-bearing formation or reservoir, it is usually possible to recover only a limited proportion of the original oil present in the reservoir. For this reason, a variety of supplemental recovery techniques have been used to improve the displacement of oil from the reservoir rock. These techniques can be generally classified as thermally based recovery methods (such as steam flooding operations), waterflooding methods, and gas-drive based methods that can be operated under either miscible or immiscible conditions.
In miscible flooding operations, an injection fluid or solvent is injected into the reservoir to form a single-phase solution with the oil in place so that the oil can then be removed as a more highly mobile phase from the reservoir. The solvent is typically a light hydrocarbon such as liquefied petroleum gas (LPG), a hydrocarbon gas containing relatively high concentrations of aliphatic hydrocarbons in the C2 to C6 range, nitrogen, or carbon dioxide. Miscible recovery operations are normally carried out by a displacement procedure in which the solvent is injected into the reservoir through an injection well to displace the oil from the reservoir towards a production well from which the oil is produced. This provides effective displacement of the oil in the areas through which the solvent flows. Unfortunately, the solvent often flows unevenly through the reservoir.
Because the solvent injected into the reservoir is typically substantially less viscous than the resident oil, the solvent often fingers and channels through the reservoir, leaving parts of the reservoir unswept. Added to this fingering is the inherent tendency of a highly mobile solvent to flow preferentially through the more permeable rock sections or to gravity override in the reservoir.
The solvent's miscibility with the reservoir oil also affects its displacement efficiency within the reservoir. Some solvents, such as LPG, mix directly with reservoir oil in all proportions and the resulting mixtures remain single phase. Such solvent is said to be miscible on first contact or “first-contact miscible.” Other solvents used for miscible flooding, such as carbon dioxide or hydrocarbon gas, form two phases when mixed directly with reservoir oil—therefore they are not first-contact miscible. However, at sufficiently high pressure, in-situ mass transfer of components between reservoir oil and solvent forms a displacing phase with a transition zone of fluid compositions that ranges from oil to solvent composition, and all compositions within the transition zone of this phase are contiguously miscible. Miscibility achieved by in-situ mass transfer of the components resulting from repeated contact of oil and solvent during the flow is called “multiple-contact” or dynamic miscibility. The pressure required to achieve multiple-contact miscibility is called the “minimum-miscibility pressure.” Solvents just below the minimum miscibility pressure, called “near-miscible” solvents, may displace oil nearly as well as miscible solvents.
Predicting miscible flood performance in a reservoir requires a realistic model representative of the reservoir. Numerical simulation of reservoir models is widely used by the petroleum industry as a method of using a computer to predict the effects of miscible displacement phenomena. In most cases, there is desire to model the transport processes occurring in the reservoir. What is being transported is typically mass, energy, momentum, or some combination thereof. By using numerical simulation, it is possible to reproduce and observe a physical phenomenon and to determine design parameters without actual laboratory experiments and field tests.
Reservoir simulation infers the behavior of a real hydrocarbon-bearing reservoir from the performance of a numerical model of that reservoir. The objective is to understand the complex chemical, physical, and fluid flow processes occurring in the reservoir sufficiently well to predict future behavior of the reservoir to maximize hydrocarbon recovery. Reservoir simulation often refers to the hydrodynamics of flow within a reservoir, but in a larger sense reservoir simulation can also refer to the total petroleum system which includes the reservoir, injection wells, production wells, surface flowlines, and surface processing facilities.
The principle of numerical simulation is to numerically solve equations describing a physical phenomenon by a computer. Such equations are generally ordinary differential equations and partial differential equations. These equations are typically solved using numerical methods such as the finite element method, the finite difference method, the finite volume method, and the like. In each of these methods, the physical system to be modeled is divided into smaller gridcells or blocks (a set of which is called a grid or mesh), and the state variables continuously changing in each gridcell are represented by sets of values for each gridcell. In the finite difference method, an original differential equation is replaced by a set of algebraic equations to express the fundamental principles of conservation of mass, energy, and/or momentum within each gridcell and transfer of mass, energy, and/or momentum transfer between gridcells. These equations can number in the millions. Such replacement of continuously changing values by a finite number of values for each gridcell is called “discretization”. In order to analyze a phenomenon changing in time, it is necessary to calculate physical quantities at discrete intervals of time called timesteps, irrespective of the continuously changing conditions as a function of time. Time-dependent modeling of the transport processes proceeds in a sequence of timesteps.
In a typical simulation of a reservoir, solution of the primary unknowns, typically pressure, phase saturations, and compositions, are sought at specific points in the domain of interest. Such points are called “gridnodes” or more commonly “nodes.” Gridcells are constructed around such nodes, and a grid is defined as a group of such gridcells. The properties such as porosity and permeability are assumed to be constant inside a gridcell. Other variables such as pressure and phase saturations are specified at the nodes. A link between two nodes is called a “connection.” Fluid flow between two nodes is typically modeled as flow along the connection between them.
Compositional modeling of hydrocarbon-bearing reservoirs is necessary for predicting processes such as first-contact miscible, multiple-contact miscible, and near-miscible gas injection. The oil and gas phases are represented by multicomponent mixtures. In such modeling, reservoir heterogeneity and viscous fingering and channeling cause variations in phase saturations and compositions to occur on scales as small as a few centimeters or less. A fine-scale model can represent the details of these adverse-mobility displacement behaviors. However, use of fine-scale models to simulate these variations is generally not practical because their fine level of detail places prohibitive demands on computational resources. Therefore, a coarse-scale model having far fewer gridcells is typically developed for reservoir simulation. Considerable research has been directed to developing models suitable for use in predicting miscible flood performance.
Development of a coarse-grid model that effectively simulates gas displacement processes is especially challenging. For compositional simulations, the upscaled, coarse-grid model must effectively characterize changes in phase behavior and changes in oil and gas compositions as the oil displacement proceeds. Many different techniques have been proposed. Most of these proposals use empirical models to represent viscous fingering in first-contact miscible displacement. See for example:                Koval, E. J., “A Method for Predicting the Performance of Unstable Miscible Displacement in Heterogeneous Media,” Society of Petroleum Engineering Journal, pages 145-154, June 1963;        Dougherty, E. L., “Mathematical Model of an Unstable Miscible Displacement,” Society of Petroleum Engineering Journal, pages 155-163, June 1963;        Todd, M. R., and Longstaff, W. J., “The Development, Testing, and Application of a Numerical Simulator for Predicting Miscible Flood Performance,” Journal of Petroleum Technology, pages 874-882, July 1972;        Fayers, F. J., “An Approximate Model with Physically Interpretable Parameters for Representing Miscible Viscous Fingering,” SPE Reservoir Engineering, pages 542-550, May 1988; and        Fayers, F. J. and Newley, T. M. J., “Detailed Validation of an Empirical Model for Viscous Fingering with Gravity Effects,” SPE Reservoir Engineering, pages 542-550, May 1988.        
Of these models, the Todd-Longstaff (“T-L”) mixing model is the most popular, and it is used widely in reservoir simulators. When properly used, the T-L mixing model provides reasonably accurate average characteristics of adverse-mobility displacements when the injected solvent and oil are first-contact miscible. However, the T-L mixing model is less accurate under multiple-contact miscible conditions.
Models have been suggested that use the T-L model to account for viscous fingering under multiple-contact miscible situations (see for example Todd, M. R. and Chase, C. A., “A Numerical Simulator for Predicting Chemical Flood Performance,” SPE-7689, presented at the 54th Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers, Houston, Tex., 1979, sometimes referred to as the “Todd-Chase technique”). In modeling a multiple-contact miscible displacement, in addition to the viscous fingering taken into account in the T-L mixing model, exchange of solvent and oil components between phases according to the phase behavior relations must also be considered. The importance of the interaction between phase behavior and fingering in multiple-contact miscible displacements was disclosed by Gardner, J. W., and Ypma, J. G. J., “An Investigation of Phase-Behavior/Macroscopic Bypassing Interaction in CO2 Flooding,” Society of Petroleum Engineering Journal, pages 508-520, October 1984. However, these proposals did not effectively combine use of a mixing model and a phase behavior model.
Another proposed model for taking into account fingering and channeling behavior in multiple-contact miscible displacement suggested making the dispersivities of solvent and oil components dependent on the viscosity gradient, thereby addressing the macroscopic effects of viscous fingering (see Young, L. C., “The Use of Dispersion Relationships to Model Adverse Mobility Ratio Miscible Displacements,” paper SPE/DOE 14899 presented at the 1986 SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, April 20-23). Another model proposed extending the T-L model to multiphase multicomponent flow with simplified phase behavior predictions (see Crump, J. G., “Detailed Simulations of the Effects of Process Parameters on Adverse Mobility Ratio Displacements,” paper SPE/DOE 17337, presented at the 1988 SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, April 17-20). A still another model suggested using the fluid compositions flowing out of a large gridcell to compensate for the nonuniformity of the fluid distribution in the gridcell (see Barker, J. W., and Fayers, F. J., “Transport Coefficients for Compositional Simulation with Coarse Grids in Heterogeneous Media”, SPE 22591, presented at SPE 66th Annual Tech. Conf., Dallas, Tex., Oct. 6-9, 1991). A still another model proposed that incomplete mixing between solvent and oil can be represented by assuming that thermodynamic equilibrium prevails only at the interface between the two phases, and a diffusion process drives the oil and solvent composition towards these equilibrium values (see Nghiem, L. X., and Sammon, P. H., “A Non-Equilibrium Equation-of-State Compositional Simulator,” SPE 37980, presented at the 1997 SPE Reservoir Simulation Symposium, Dallas, Tex., Jun. 8-17, 1997). The gridcells in these models were not subdivided.
Proposals have been made to represent fingering and channeling in multiple-contact miscible displacements using two-region models. See for example:                Nghiem, L. X., Li, Y. K. and Agarwal, R. K., “A Method for Modeling Incomplete Mixing in Compositional Simulation of Unstable Displacements,” SPE 18439, presented at the 1989 Reservoir Simulation Symposium, Houston, Tex., Feb. 6-8, 1989; and        Fayers, F. J., Barker, J. W., and Newley, T. M. J., “Effects of Heterogeneities on Phase Behavior in Enhanced Oil Recovery,” in The Mathematics of Oil Recovery, P. R. King, editor, pages 115-150, Clarendon Press, Oxford, 1992.These models divide a simulation gridcell into a region where complete mixing occurs between the injected solvent and a portion of the resident oil and a region where the resident oil is bypassed and not contacted by the solvent. Although the conceptual structure of these models appears to provide a better representation of incomplete mixing in multiple-contact miscible displacements than single zone models, the physical basis of the equations used to represent bypassing and mixing is unclear. In particular, these models (1) use empirical correlations to represent oil/solvent mobilities in each region, (2) use empirical correlations to represent component transfer between regions, and (3) make restrictive assumptions about the composition of the regions and direction of component transfer between the regions. It has been suggested that the empirical mobility and mass transfer functions in these models can be determined by fitting them to the results of fine-grid simulations. As a result, in practice, calibration of these models will be a time-consuming and expensive process. Furthermore, these models are unlikely to accurately predict performance outside the parameter ranges explored in the reference fine-grid simulations.        
While the two-region approaches proposed in the past have certain advantages, there is a continuing need for improved simulation models that provide a better physical representation of bypassing and mixing in adverse mobility displacement and thus enable more accurate and efficient prediction of flood performance.